Autocovariance Structure of Markov Regime Switching Models and Model Selection

We show that the covariance function of a second-order stationary vector Markov regime switching time series has a vector ARMA(p,q) representation, where upper bounds for p and q are elementary functions of the number of regimes. These bounds apply to vector Markov regime switching processes with both mean–variance and autoregressive switching. This result yields an easily computed method for setting a lower bound on the number of underlying Markov regimes from an estimated autocovariance function.